0.2 A multiresolution formulation of wavelet systems  (Page 5/8)

It is important at this point to recognize the relationship of the scaling function part of the expansion [link] to the wavelet part of the expansion. From the representation of the nested spaces in [link] we see that the scaling function can be defined at any scale $j$ . [link] uses $j=0$ to denote the family of scaling functions.

You may want to examine the Haar system example at the end of this chapter just now to see these features illustrated.

The discrete wavelet transform

Since

using [link] and [link] , a more general statement of the expansion [link] can be given by

or

where $j_0$ could be zero as in [link] and [link] , it could be ten as in [link] , or it could be negative infinity as in [link] and [link] where no scaling functions are used. The choice of $j_0$ sets the coarsest scale whose space is spanned by ${\phi }_{j_0,k}\left(t\right)$ . The rest of $L^2\left(\mathbf{R}\right)$ is spanned by the wavelets which provide the high resolution details of the signal. In practice where one is given only thesamples of a signal, not the signal itself, there is a highest resolution when the finest scale is the sample level.

The coefficients in this wavelet expansion are called the discrete wavelet transform (DWT) of the signal $g\left(t\right)$ . If certain conditions described later are satisfied, these wavelet coefficientscompletely describe the original signal and can be used in a way similar to Fourier series coefficients for analysis, description, approximation, andfiltering. If the wavelet system is orthogonal, these coefficients can be calculated by inner products

and

If the scaling function is well-behaved, then at a high scale, the scaling is similar to a Dirac delta function and the inner product simply samples thefunction. In other words, at high enough resolution, samples of the signal are very close to the scaling coefficients. More is said about this later.It has been shown [link] that wavelet systems form an unconditional basis for a large class of signals. That is discussed in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients but means that even for the worst case signal in the class, the wavelet expansion coefficients drop off rapidly as $j$ and $k$ increase. This is why the DWT is efficient for signal and image compression.

The DWT is similar to a Fourier series but, in many ways, is much more flexible and informative. It can be made periodic like a Fourier seriesto represent periodic signals efficiently. However, unlike a Fourier series, it can be used directly on non-periodic transient signals withexcellent results. An example of the DWT of a pulse was illustrated in Figure: Two-Stage Two-Band Analysis Tree . Other examples are illustrated just after the next section.

A parseval's theorem

If the scaling functions and wavelets form an orthonormal basis or a tight frame defined in Chapter: Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Right Frames, and unconditional Bases , there is a Parseval's theorem that relates the energy of the signal $g\left(t\right)$ to the energy in each of the components and their wavelet coefficients. That isone reason why orthonormality is important.